US7454690B1 - Architecture and control of reed-solomon list decoding - Google Patents
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Abstract
Description
for a modified decision-codeword can be computed by using the computations below:
Λ(e
B (e
Λ(e
B (e
where the variables and their computations are described by the Burd reference.
The upper bound of the summations are not specified to indicate that the degree of the polynomials in the equations can vary. Also, the intermediate variable D can be computed by:
If D=0, then the new error indicators cannot be computed using iterative decoding, and another technique such as the BMA algorithm will need to be used. On the other hand, if D is not equal to 0, then variables a, b, and c can be computed and are given by:
The computed values a, b, and c, can then be used in equations EQ3 and EQ4 to compute the new error locator polynomial Λ(e
can be computed in one embodiment by dividing Λ(e
and when there are A extra syndromes, the validity test is:
One of equations EQ10 and EQ11 is used depending on the number of extra syndromes. When the equality or equalities are determined to be true, the new error locator polynomial can be presumed to be valid and can be passed to a Chien search module where it's validity can be verified. Alternatively, the validity test of equations EQ10 and EQ11 can be modified to use the modified syndromes and {circumflex over (Λ)}(e
and when there are Δ extra syndromes, the validity test becomes:
Equations EQ12 and EQ13 operate in the same way as equations EQ10 and EQ11 so that when the equality or equalities are determined to be true, the new error locator polynomial can be presumed to be valid.
For case one where deg(Λ(x))<deg(B(x))+2, the values d1 and d3 are computed as shown in equation EQ14, but the value d2 is computed instead of d4 by:
By using the lower syndromes, equations EQ10-EQ11 are replaced by:
where v is the degree of {circumflex over (Λ)}t (e
-
- (1) At least a few incidence vectors having one error event should be ordered first, but not all single-error incidence vectors need to be computed first;
- (2) Among incidence vectors having the same number of error events, incidence vectors that do not need to be used in any subsequent iterative decoding iteration should be ordered earlier;
- (3) Incidence vectors whose child combinations require least storage should be ordered earlier;
- (4) If incidence vectors are processed in a pipeline architecture and each incidence vector requires k pipeline stages, then any children of an incidence vector must be ordered at least k time slots later than the parent; and
- (5) Achieve a substantially similar number of child incidence vectors for each parent incidence vector.
Since the results of incidence vectors that have no children are not needed for subsequent iterative decoding iterations, ordering incidence vectors based on guideline (2) can reduce the amount of memory needed for iterative decoding. In one embodiment, each incidence vector can be associated with an indicator that indicates whether the error indicators associated with an incidence vector need to be stored in memory.
TABLE 1 | |
Order | Incidence vector |
1 | (e1)r1 |
2 | (e2)r2 |
3 | (e4)r3 |
4 | (e1 | e2)r1 |
5 | (e2 | e3)r2 |
6 | [e4 | e2] |
7 | [e4 | e3] |
8 | (e4 | e1)r3 |
9 | [el + e2 | e6] |
10 | [e4 + e1 | e2] |
11 | [e4 + e1 | e3] |
12 | (e3)r3 |
13 | [e1 + e2 | e3] |
14 | (e3 | e1)r3 |
15 | (el + e2 | e5)r1 |
16 | [e2 + e3 | e6] |
17 | [el + e2 + e5 | e3] |
. | |
. | |
. | |
Table one shows an ordering of seventeen incidence vectors out of a total of sixty-three. The particular ordering and memory arrangement are exemplary and other variations are contemplated.
S i (e
where l is the number of the error event that is corrected, and i corresponds to a particular syndrome. For an exemplary error event l=2, the syndromes are
S 0 (e
S 1 (e
S 2 (e
S 3 (e
.
.
.
and so on. The previously computed syndromes Si, the error value e2, symbol location j2, and primitive element α can be stored in a soft-information component memory. Thus, the only quantities that need to be computed are the terms e2αi·j
and
which provide quotient polynomials QΛ(x) and QB(x), respectively, and remainder values rA and rB, respectively. The remainders are numerical values rather than polynomials because the divisor is a first degree polynomial. These results can be used to obtain the other quotients and remainders as follows:
Q xΛ(x)=xQ Λ(x)+r Λ (EQ19)
r xΛ =r Λα−j
Q xB(x)=xQ B(x)+r B, (EQ21)
r xB =r Bα−j
Q x
r x
P(x)=p 0 x t +p 1 x t−1 +p 2 x t−2 + . . . +p t−1 x+p t
by (x+α−j). Let the quotient polynomial be
Q(x)=q 0 x t−1 +q 1 x x−2 + . . . +q t−2 x+q t−1,
and let r be the remainder. Because the divisor polynomial (x+α−j) has degree one, the relationship between the polynomials becomes P(x)=Q(x)x+Q(x)α−j+r. This means that the following recursion equation can be used to determine the coefficients of Q(x) and r:
q i =p i +q i−1α−j, for 1≦i≦t−1, (EQ25)
where initially q0=p0, and the remainder is r=pt+qt−1α−j. The
q 1 =p 1 +p 0α−j, (EQ26)
q 2 =p 2 +p 1α−j +p 0α−2j, (EQ27)
q 3 =p 3 +p 2α−j +p 1α−2j +p 0α−3j, (EQ28)
q 4 =p 4 +p 3α−j +p 2α−2j +p 1α−3j +p 0α−4j, (EQ29)
q 5 =p 5 +p 4α−j +p 3α−2j +p 2α−3j +p 1α−4j +p 0α−5j, (EQ30)
q 6 =p 6 +p 5α−j +p 4α−2j +p 3α−3j +p 2α−4j +p 1α−5j +p 0α−6j. (EQ31)
In this example, the implementation allows six quotient coefficients to be computed in parallel, but still requires recursion to compute quotient coefficients beyond the initial six. However, each subsequent iteration can also compute six coefficients in parallel. Specifically, after q1, . . . , q6 are computed, the next iteration can compute q7, . . . , q12 by setting p0 equal to q6 and replacing p1, . . . , p6 with p7, . . . , p12, respectively. In this manner, the number of iterations needed to compute quotient coefficients in a long division operation can be reduced by six.
Step | Evaluation |
0 | D = rxΛd3 + rxBd2 | (intermediate result) |
a = rΛd3 + rxBd1 | (intermediate result) | |
1 | | (intermediate result) |
D−1 = (D)−1 | (D complete) | |
2 | a = a(D−1) | (a complete) |
| (intermediate result) | |
3 | | (b complete) |
The number of steps and the particular progression of sub-computations in the table above are exemplary and other variations are contemplated. Additionally, values and variables other than those shown can also be performed by interleaved sub-computations that are arranged in a progressive, step-wise manner.
over a computation interval. Assume, as before, that t=36, and also assume that the degree of an error locator polynomial Λ(x) is less than or equal to t. The first error locator polynomial coefficient Λ36 can be pre-loaded into the delay memory component 2014 (
can be performed in the same manner. The computation for coefficients for other quotient polynomials QxΛ(x), QxB(x), and Qx
Claims (19)
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US12/541,720 US8132083B1 (en) | 2004-10-27 | 2009-08-14 | Architecture and control of reed-solomon list decoding |
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Cited By (8)
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US8527850B1 (en) | 2013-09-03 |
US20090063937A1 (en) | 2009-03-05 |
US8788916B2 (en) | 2014-07-22 |
US20090292976A1 (en) | 2009-11-26 |
US20090083608A1 (en) | 2009-03-26 |
US8245118B2 (en) | 2012-08-14 |
US20120137197A1 (en) | 2012-05-31 |
US7590924B2 (en) | 2009-09-15 |
US8296635B2 (en) | 2012-10-23 |
US20120284588A1 (en) | 2012-11-08 |
US7444582B1 (en) | 2008-10-28 |
US20090055717A1 (en) | 2009-02-26 |
US7590923B1 (en) | 2009-09-15 |
US8635513B1 (en) | 2014-01-21 |
US8219894B2 (en) | 2012-07-10 |
US8132082B2 (en) | 2012-03-06 |
US8132083B1 (en) | 2012-03-06 |
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